These questions ask you to evaluate expressions where negative signs make the usual order of operations easy to lose. You need to choose the next operation, preserve each intermediate sign and continue until the entire line is finished.
Reduce one layer at a time
Mark the deepest brackets first and replace only that part with its value. Next handle powers. Then scan from left to right for multiplication and division, which share one rank. Finally scan from left to right for addition and subtraction, which also share one rank. For example, 7 + (-3) times 4 becomes 7 + (-12), then -5. The addition waits because multiplication has the higher rank.
| Signal | Next move |
|---|---|
| Brackets inside brackets | Complete the innermost group, replace it with one signed value, then work outward. |
| Multiplication beside division | They have equal rank, so use whichever occurs first from the left. |
| Addition beside subtraction | They have equal rank, so work from left to right. |
| Subtracting a negative value | Rewrite it as adding the positive opposite before continuing. |
| A power involving a negative sign | Check whether the sign is inside the repeated factor or outside the grouped power. |
Keep a short line of working so each sign has a visible owner. In 18 divided by (-3) times 2, the division occurs first and gives -6, then multiplication gives -12. For a bracket example, 3 times (2 - (-5)) first becomes 3 times 7, then 21. For powers, negative three squared means (-3) times (-3), which is 9. A minus outside the grouped square waits: the negative of three squared is -9.
Order and sign traps
- Ignoring brackets lets an outside operation enter too early. Finish the whole grouped sum before using the factor or divisor beside it.
- Stopping after a bracket gives only an intermediate value. Check the expression on both sides of the reduced group before finishing.
- Doing multiplication before an earlier division invents a rank difference that does not exist. Equal-rank operations run from left to right.
- Reading subtraction of a negative as one ordinary subtraction moves in the wrong direction. Replace the pair with addition of the opposite.
- Moving a minus sign into or out of parentheses changes what is multiplied in a square or cube. Count exactly which signed factors the grouping contains.
- Distributing a factor to only the first term of a grouped sum leaves part of the group untouched. Complete the bracket first unless you deliberately distribute to every term.
Try it
Work out 3 + (-2) * 5. Enter the value.
The distance from -4 to 3 on a number line is written as 3 - (-4), and that distance is then doubled, giving 2 * (3 - (-4)). What is the value?
- -2
- 10
- 7
- 14
Work out (-18 + 6) / (-4). Enter the value.